We investigate the ionization of icosahedral fullerenes (C-20, C-60, C-80, and C-180) in an intense laser pulse using the S-matrix theory. The results obtained are in excellent agreement with the recent observations of unexpectedly high saturation intensities of the Buckminster fullerene and its multiply charged ions. Our analysis strongly suggests that the related phenomenon of suppressed ionization of these complex fullerenes is due to the finite cage size and the "multislit" interference effect between partial waves emitted from the different nuclei rather than to a dynamical multielectron polarization effect.